complex singularity exponent, lct

Question

Hi everybody,

I have a question about log canonical thresholds / complex singularity exponents.

If I understood well, this invariant sees more things than the multiplicity, for example, the cusp in dimension 1 ($x^3+y^2=0$) has $c_0 = 5/6$ and the ordinary quadratic singularity has c_0=1.

I have several questions :

1) Is it true that for the singularity $x^a+y^b+z^c=0$, the complex singulary exponent is $min(1;1/a+1/b+1/c)$ ?

2) If it's true, how to distinguish $x^3+y^2+z^2=0$ and the ordinary quadratic surface singularity ? Is there another invariant (besides the Milnor number) ?

3) If I have a variety with isolated singularities, does it make sense to try to measure its 'singularity' by adding the exponents at all the singular points ? Or taking the inf ?

4) compared to the Milnor number, what is the advantage of working with this invariant ? I thought it would be a way to say that a curve with a cusp is 'more singular' than a curve with two ordinary quadratic points (with the Milnor number they are 'equaly singular'), but I don't know if that makes sense.

Thanks in advance,

J-B B

Answer 1

With regards to your questions.

1) Yup, see Example 9.3.31 and Theorem 9.3.37 in Lazarsfeld's book, "Positivity in Algebraic Geometry II".

2) You might not be aware of things like (minimal) log discrepancies. Of course there are various properties of the minimal resolutions too.

3) Do you mean exponents of different variables or the same variables, what did you have in mind? I don't know of any good geometric interpretation of doing this in general.

4) This log canonical threshold also appears in many other unexpected contexts? See for example:

Budur's initial question in "Singularity invariants related to Milnor fibers: survey". http://arxiv.org/pdf/1012.3150